In an integral value measuring circuit including a capacitor and an operational amplifier, the capacitor is connected between input and output sides of the operational amplifier as illustrated in FIG. 3.
In the circuit illustrated in FIG. 3, the following relationship is established:i=Cp(Vo−Vi′),
where Vi denotes an input voltage to the operational amplifier, Vi′ denotes a voltage of the negative input terminal of the operational amplifier, Vo denotes an output voltage, R denotes a resistance value of the input side, C denotes a capacitance of the capacitor, i denotes an input current value in the operational amplifier, and p denotes a differentiation operator.
Meanwhile, considering (Vi′−Vi)=iR,i=Cp(Vo−Ri−Vi).
On the other hand Vo=−AVi′=−A(iR+Vi) wherein A denotes the degree of amplification of amplifier 1 in FIG. 3.
So, this leads to the following equation by deleting i
      V    o    =                              -                      V            i                          /                  (                      CRp            +                          CRp              /              A                        -                          1              /              A                                )                    ≈              -                  (                                    V              i                        /            CRp                    )                      =                  -                  (                      1            /            CR                    )                    ⁢                        ∫          o          t                ⁢                              v            i                    ⁢                                          ⁢                      ⅆ            t                              because the value of A is far larger than 1.
That is, since the voltage Vo in the output side of the operational amplifier is proportional to the integral value of the input voltage Vi, it is possible to measure the integral value of the input voltage Vi.
In the case of the integral value measurement circuit in prior art, as illustrated in FIG. 3, a switch is provided in parallel between both electrode sides of the capacitor. The capacitor is charged by turning on or off the separate switch SW, so that an integral state is formed as indicated by the aforementioned Equation 1.
However, a leakage resistance element and a parasitic capacitance are formed when the separate switch SW changes from the ON state to the OFF state, so that a noise current flows by itself. Therefore, it is difficult to obtain a simple integral equation in the aforementioned Equation 1.
The aforementioned difficulty will be described based on numerical formulas. Since the leakage resistance and the parasitic capacitance are connected in series (this fact is apparent because the noise current decays exponentially when the switch changes from the ON state to the OFF state), the following relationship is established between the input voltage Vi of the operational amplifier and the output voltage Vo:Vi/R≈−{Cp+1/(r+1/c′p)}Vo=−{Cp+c′p/(1+c′rp)}Vo 
where r denotes the leakage resistance value of the capacitor, and c′ denotes the parasitic capacitance value. Furthermore, this fact may become apparent from the following complicated differentiation and integral formulas:Vo≈−(1+c′rp)Vi/R·(Cp+c′p+Cc′rp2)    [Non-patent Literature 1] “Practical Electronic Circuit Design Guide,” Takashi KENJO, et al, published by SOGO Electronic Publication Corporation in Japan, April 1981